Question

Determine (a) the equation of lowest degree with rational coefficients, one of whose roots is \(\sqrt{13}-\sqrt{3}\); (b) the equation of lowest degree with real or complex coefficients having the roots 4 and \(4-9 \mathrm{i}\).

Short answer

The short answers for both parts are: (a) The equation of lowest degree with rational coefficients having the root \(\sqrt{13}-\sqrt{3}\) is: \[P(x) = x^2 - 2\sqrt{13}x + 10\] (b) The equation of lowest degree with real or complex coefficients having the roots 4 and \(4-9i\) is: \[Q(x) = (x - 4)(x^2 - 8x + 97)\]

Step-by-step solution

Identify the root

Given root is \(\sqrt{13}-\sqrt{3}\).

Find the conjugate root

The conjugate of the given root is \(\sqrt{13}+\sqrt{3}\).

Write the equation

Let the polynomial equation be P(x). Since \(P(\sqrt{13}-\sqrt{3}) = 0\) and \(P(\sqrt{13}+\sqrt{3}) = 0\), we can write the equation as: \[P(x) = (x - (\sqrt{13}-\sqrt{3}))(x - (\sqrt{13}+\sqrt{3}))\]

Simplify the equation

Now, multiply the factors to get the polynomial equation: \begin{align*} P(x) &= (x - (\sqrt{13}-\sqrt{3}))(x - (\sqrt{13}+\sqrt{3})) \\ &= (x - \sqrt{13} + \sqrt{3})(x - \sqrt{13} - \sqrt{3}) \\ &= x^2 - \sqrt{13}x - \sqrt{3}x + \sqrt{13}x + 13 + \sqrt{39} + \sqrt{3}x - \sqrt{39} - 3 \\ &= x^2 - 2\sqrt{13}x + 10. \end{align*} So, the equation of lowest degree with rational coefficients having the root \(\sqrt{13}-\sqrt{3}\) is: \[P(x) = x^2 - 2\sqrt{13}x + 10\] ### Part (b) ###

Identify the roots

Given roots are 4 and \(4-9i\).

Find the conjugate root

The conjugate of the given complex root is \(4+9i\).

Write the equation

Let the polynomial equation be Q(x). Since \(Q(4) = 0\), \(Q(4-9i) = 0\), and \(Q(4+9i) = 0\), we can write the equation as: \[Q(x) = (x - 4)(x - (4-9i))(x - (4+9i))\]

Simplify the equation

First, multiply the complex factors and then the real factor to get the polynomial equation: \begin{align*} Q(x) &= (x - 4)(x - (4-9i))(x - (4+9i)) \\ &= (x - 4) [(x - 4 + 9i)(x - 4 - 9i)] \\ &= (x - 4)(x^2 - 8x + 16 + 81) \\ &= (x - 4)(x^2 - 8x + 97). \end{align*} So, the equation of lowest degree with real or complex coefficients having the roots 4 and \(4-9i\) is: \[Q(x) = (x - 4)(x^2 - 8x + 97)\]

Key concepts

Conjugate Roots

When you deal with polynomial equations, especially with non-real numbers, understanding the concept of conjugate roots is crucial. In mathematics, when a polynomial has rational coefficients, and a non-real complex root is present, its conjugate pair must also appear as a root. This is essential because it keeps the polynomial equation's coefficients rational.
Let's consider two cases:

• If you have a complex root like \( a + bi \), its conjugate would be \( a - bi \).
• If the root is given by square roots of irrational numbers such as \( \sqrt{m} - \sqrt{n} \), its conjugate will be \( \sqrt{m} + \sqrt{n} \).

These conjugate pairs ensure the preservation of rational coefficients in the polynomial. As a result, any non-real element is counterbalanced by its conjugate, maintaining a balance in the polynomial equation.
Hence, understanding and using the conjugate roots is necessary for constructing accurate polynomial equations that include complex or irrational roots.

Complex Numbers

In mathematics, complex numbers are a category of numbers that extend the concept of one-dimensional number lines to a two-dimensional number plane. A complex number is defined in the form \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part with \( i \) representing \( \sqrt{-1} \). Complex numbers are fundamental in solving equations that do not have real solutions.
They are vital in constructing polynomials with certain properties, like specified roots. Here's why they matter:

• Enable the resolution of equations that fail to yield real solutions.
• Allow examination of systems and phenomena in fields such as engineering and physics, where real-only numbers are insufficient.
• Facilitate the computation of polynomials by ensuring the principle of conjugate pairs, as seen in our example, where if a polynomial has a complex root, its conjugate must also be included to keep coefficients rational.

Studying complex numbers is crucial in advanced fields of study, but even basic polynomial equations benefit from understanding their role and contribution.

Rational Coefficients

Rational coefficients are coefficients in polynomial equations that are rational numbers, meaning they can be expressed as fractions of integers. When creating polynomial equations from given roots, it's crucial to ensure that the coefficients remain rational if specified. For a polynomial equation like \( P(x) \) or \( Q(x) \), rational coefficients guarantee that the polynomial has a consistent and predictable form, especially important when there are irrational or complex roots.
Why are rational coefficients essential?

• They ensure the polynomial has a real representation, even when dealing with non-real roots.
• They maintain uniformity for polynomials derived from real-world applications, where irrational coefficients are often not feasible.
• They align with the mathematical tradition, simplifying calculation and theory.

In polynomial construction, achieving rational coefficients often involves ensuring conjugate roots are paired, as seen in our example. This process prevents irrational or imaginary roots from making the equation too complex or difficult to use in practical applications.